Parametric instability of a driven ion-acoustic wave

The stability of a driven coherent ion-acoustic wave (IAW) with regard to its decay into longer wavelengths is investigated. The effects of harmonic generation and wave dispersion are taken into account. The driver frequency and wave number do not necessarily satisfy the linear ion-acoustic wave dispersion relation, allowing a frequency mismatch between the driver frequency and the plasma linear-response frequency. The stability analysis is generally shown to involve a seven wave coupling in which the fundamental and the second-harmonic components of the equilibrium IAW couple to a low-frequency daughter wave and to their Stokes and anti-Stokes satellites. The general dispersion relation corresponding to this seven wave coupling is derived and solved numerically in three-dimensional geometry. A variety of branches of instabilities is found to appear. In order to classify these branches, an approximate dispersion relation is derived, corresponding to the Korteweg-de-Vries limit for the plasma low-frequency nonlinear behavior. This approximate dispersion relation makes it possible to obtain approximate expressions for the growth rate of each branch. The occurrence of each type of instability is discussed as a function of the sign and size of the frequency mismatch. The special case of one-dimensional geometry is also considered, making it possible to investigate the results of one-dimensional numerical simulations. (C) 2005 American Institute of Physics.